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%I #15 Mar 12 2021 22:24:48
%S 1,0,-1,-2,-1,2,0,2,0,0,1,0,2,-2,-1,0,-2,-2,0,0,0,0,2,0,-1,0,2,0,0,2,
%T -1,2,0,2,0,0,-2,-2,0,0,0,-2,-2,0,1,0,0,-2,2,0,-2,2,-1,0,0,-2,2,2,2,0,
%U 0,0,2,0,2,0,0,0,0,0,-1,-2,-2,2,0,-2,0,2,-2
%N Expansion of f(-x^2) * phi(-x^3) in powers of x where phi(), f() are Ramanujan theta functions.
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H G. C. Greubel, <a href="/A258196/b258196.txt">Table of n, a(n) for n = 0..2500</a>
%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F Expansion of q^(-1/12) * eta(q^2) * eta(q^3)^2 / eta(q^6) in powers of q.
%F Euler transform of period 6 sequence [ 0, -1, -2, -1, 0, -2, ...].
%F G.f.: Product_{k>0} (1 - x^(2*k)) * (1 - x^(3*k)) / (1 + x^(3*k)).
%F a(49*n + 18) = a(49*n + 25) = a(49*n + 32) = a(49*n + 39) = a(49*n + 46) = 0.
%e G.f. = 1 - x^2 - 2*x^3 - x^4 + 2*x^5 + 2*x^7 + x^10 + 2*x^12 - 2*x^13 + ...
%e G.f. = q - q^25 - 2*q^37 - q^49 + 2*q^61 + 2*q^85 + q^121 + 2*q^145 + ...
%t a[ n_] := SeriesCoefficient[ QPochhammer[ x^2] EllipticTheta[ 4, 0, x^3], {x, 0, n}];
%o (PARI) {a(n) = if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^3 + A)^2 / eta(x^6 + A), n))};
%o (PARI) {a(n) = my(A, p, e, x, w=quadgen(-8)); if( n<0, 0, n = 12*n + 1; A = factor(n); simplify( prod( k=1, matsize(A)[1], [p, e] = A[k,]; if( p<5, 0, p%12 == 11, !(e%2), p%12 == 5, forstep(y = 3, sqrtint(2*p), 6, if( issquare(2*p - y^2, &x), if( x%6==5, x=-x); x = (x-1)/6; break)); (-1)^(e*x) * [1, w, -1, 0, -1, -w, 1, 0][e%8+1], x=0; forstep(y = 0, sqrtint(p), 6, if( issquare(p - y^2, &x), if( x%6==5, x=-x); x = [x-1, y]/6; break)); if( x==0, (-1)^(e\2) * !(e%2), (-1)^(e*(x[1] + x[2])) * (e+1))))))};
%o (PARI) q='q+O('q^99); Vec(eta(q^2)*eta(q^3)^2/eta(q^6)) \\ _Altug Alkan_, Aug 02 2018
%Y Cf. A204770.
%K sign
%O 0,4
%A _Michael Somos_, May 23 2015
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